Bi-Cubic Surface Patches
Copyright (c) Susan Laflin. August 1999.
The references for this topic are:
Rogers & Adams chapter 6 section 6.10.
Faux & Pratt section 7.2
When we come to require the tangents to join up smoothly as well as the values of the surface, it becomes necessary to move to a Bi-Cubic surface. The equation of this is given below, where the blending functions f1, f2, g1 and g2 are cubics in u or v and the following notation is used.
r(u,v) is the value of the surface at the point (u,v).
ru(u,v) is the value of dr/du at the point (u,v) and
rv(u,v) is the value of dr/dv at the point (u,v).
ruv(u,v) is the value of d2r/dudv at the point (u,v).
Bi-cubic Surface Patch.
These two `tangent vectors' are evaluated at each of the four corners. d2r/dudv is called the `twist vector'. The twist vector of a planar surface is zero, but if the four corners are not co-planar, then all surface patches, including the simple bi-linear surface, will have a twist in them. The size of this twist is given by the magnitude of the twist vector.
The equation of a bi-cubic surface may take the form: r(u,v) = [f1(u) f2(u) g1(u) g2(u)] R f1(v) f2(v) g1(v) g2(v) where R= r(0,0) r(0,1) rv(0,0) rv(0,1) r(1,0) r(1,1) rv(1,0) rv(1,1) ru(0,0) ru(0,1) ruv(0,0) ruv(0,1) ru(1,0) ru(1,1) ruv(1,0) ruv(1,1) and the functions are given by: f1(u) = 2u3 - 3u2 + 1 f2(u) = -2u3 + 3u2 g1(u) = u3 - 2u2 + u g2(u) = u3 - u2
These blending curves have the same form for both u and v. A full discussion of this patch may be found in the references.